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<p>This immediately raises the question of why to use such a procedure. In fact the reason is strongly motivated by real engineering problems. There, typically we encounter models for the dynamics of phenomena which depend on rates of change of functions, e.g. velocities and accelerations of particles or points on rigid bodies, which prompts the use of ordinary differential equations (ODEs). We can use ordinary calculus to solve ODEs, provided that the functions are nicely behaved—which means continuous and with continuous derivatives. Unfortunately, there is much interest in engineering dynamical problems involving functions that input step change or spike impulses to systems—playing pool is one example. Now, there is an easy way to smooth out discontinuities in functions of time: simply take an average value over all time. But an ordinary average will replace the function by a constant, so we use a kind of moving average which takes continuous averages over all possible intervals of <span class="process-math">\(t.\)</span> This very neatly deals with the discontinuities by encoding them as a smooth function of interval length <span class="process-math">\(s.\)</span></p>
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